Mathematics for the interested outsider

Line Integrals

We now define some particular kinds of integrals as special cases of our theory of integrals over manifolds. And the first such special case is that of a line integral.

Consider an oriented curve in the manifold . We know that this is a singular-cube, and so we can pair it off with a -form . Specifically, we pull back to on the interval and integrate.

More explicitly, the pullback is evaluated as

That is, for a , we take the value of the -form at the point and the tangent vector and pair them off. This gives us a real-valued function which we can integrate over the interval.

So, why do we care about this particularly? In the presence of a metric, we have an equivalence between -forms and vector fields . And specifically we know that the pairing is equal to the inner product — this is how the equivalence is defined, after all. And thus the line integral looks like

Often the inner product is written with a dot — usually called the “dot product” of vectors — in which case this takes the form

We also often write as a “vector differential-valued function”, in which case we can write

Of course, we often parameterize a curve by a more general interval than , in which case we write

This expression may look familiar from multivariable calculus, where we first defined line integrals. We can now see how this definition is a special case of a much more general construction.

Share this:

Like this:

Related

[…] any -form on the image of — in particular, given an defined on — we can define the line integral of over . We already have a way of evaluating line integrals: pull the -form back to the parameter […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.